Find fractions between two on the number line

3.NF.A.23.NF.A.3 · take · grade 3

Archetype: Compare Fractions and Decimals by Structure · step in a 7-type progression

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{5}{12}$ and $\dfrac{3}{4}$ .

$\dfrac{5}{6} \qquad \dfrac{11}{12} \qquad \dfrac{1}{2} \qquad \dfrac{4}{6}$

(figure) A number line from 0 to 1 is divided into 12 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{5}{6}$ , $\dfrac{11}{12}$ , $\dfrac{1}{2}$ , and $\dfrac{4}{6}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 12 equal parts, mark the fractions 5/6, 11/12, 1/2, and 4/6, then list every one of them that lies strictly between 5/12 and 3/4.

Givens

The number line from 0 to 1 is split into 12 equal parts, so each tick is 1/12
The four fractions to place are 5/6, 11/12, 1/2, and 4/6
The target interval is between 5/12 and 3/4

Unknowns

Which of the four fractions fall between 5/12 and 3/4

Constraints

'Between' means greater than 5/12 and less than 3/4
Comparisons are easiest when all fractions share the denominator 12

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 12 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 12 equal parts, write the boundaries over 12: 5/12 stays 5/12, and 3/4 = 9/12. So we want fractions between 5/12 and 9/12.

\dfrac{3}{4} = \dfrac{9}{12}

Matching the tick spacing makes every position a count of twelfths.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 12: 5/6 = 10/12, 11/12 stays 11/12, 1/2 = 6/12, and 4/6 = 8/12.

\dfrac{5}{6}=\dfrac{10}{12},\;\; \dfrac{1}{2}=\dfrac{6}{12},\;\; \dfrac{4}{6}=\dfrac{8}{12}

Equivalent fractions over 12 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in twelfths to the range (5/12, 9/12): 6/12 (=1/2) yes, 8/12 (=4/6) yes, 10/12 (=5/6) no (too big), 11/12 no (too big). So 1/2 and 4/6 lie between.

\dfrac{5}{12} < \dfrac{6}{12},\ \dfrac{8}{12} < \dfrac{9}{12} \le \dfrac{10}{12},\dfrac{11}{12}

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 4/6

Review

In twelfths the target window is 6/12 through 8/12; 1/2 = 6/12 and 4/6 = 8/12 sit inside it, while 5/6 = 10/12 and 11/12 sit past 9/12, so they are correctly excluded. The marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place 5/12 and 9/12, then read which of the four marked points land between them, giving the same 1/2 and 4/6.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting 5/6, 4/6, 1/2, and 3/4 as equivalent twelfths to compare

💡 Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!